I would like to start using TestLink on my work laptop as sort of a demo to see if it suits my needs. I downloaded TestLink and started the installation, and I cancelled it when it asked about opening port 80. Do all these bitnami applications need to be run on a dedicated piece of hardware running a web server? Or can I just install this on my work laptop and not get fired for ruining the whole company's network? How does all this work? Sorry for the bad question, but I seem to not know enough to even ask a good question.

You are being asked about the port 80 because there is already another application that is taken that port (it could be another bitnami stack or other program like skype for example). That is the default port for the apache web server. You could also install the stack on other different port (81 for example) and then access to the application specifying the port number: